\documentclass[9pt]{article}
\usepackage{amsmath}
\begin{document}
Read in the following dictionary:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{17}   &  -3.0 & -6.00 x_{1} & -5.00 x_{2} & +  2.00 x_{3} & -7.00 x_{4} & -6.00 x_{5} & -8.00 x_{6} &   & -5.00 x_{8} & +  2.00 x_{9} & +  5.00 x_{10} & +  3.00 x_{11} & + 10.00 x_{12} & +  2.00 x_{13} & -2.00 x_{14} & -9.00 x_{15} & -5.00 x_{16}\\
 x_{18}   &  3.0 & +  1.00 x_{1} & -10.00 x_{2} & -9.00 x_{3} & +  5.00 x_{4} & +  9.00 x_{5} & -5.00 x_{6} & -1.00 x_{7} & +  8.00 x_{8} & +  7.00 x_{9} & +  5.00 x_{10} & -7.00 x_{11} & +  6.00 x_{12} & +  4.00 x_{13} & -6.00 x_{14} & +  8.00 x_{15} & +  3.00 x_{16}\\
 x_{19}   &  -1.0 & +  5.00 x_{1} & -4.00 x_{2} & +  7.00 x_{3} & +  1.00 x_{4} & -5.00 x_{5} & +  3.00 x_{6} & +  6.00 x_{7} & +  3.00 x_{8} & -6.00 x_{9} & +  3.00 x_{10} & -4.00 x_{11} & -9.00 x_{12} & -9.00 x_{13} & +  1.00 x_{14} & +  8.00 x_{15} & +  9.00 x_{16}\\
 x_{20}   &  0.0 & +  7.00 x_{1} & -3.00 x_{2} & +  3.00 x_{3} & +  8.00 x_{4} &   & +  5.00 x_{6} & -4.00 x_{7} & +  1.00 x_{8} & +  1.00 x_{9} & +  5.00 x_{10} & -8.00 x_{11} & + 10.00 x_{12} & +  8.00 x_{13} & -9.00 x_{14} & +  8.00 x_{15} & +  1.00 x_{16}\\
 x_{21}   &  3.0 & +  6.00 x_{1} & -2.00 x_{2} &   & +  4.00 x_{4} & +  4.00 x_{5} & -8.00 x_{6} &   & -7.00 x_{8} & -3.00 x_{9} & -9.00 x_{10} & -1.00 x_{11} & +  3.00 x_{12} & -6.00 x_{13} &   & +  3.00 x_{15} & +  9.00 x_{16}\\
 x_{22}   &  -3.0 & -7.00 x_{1} & -3.00 x_{2} & -8.00 x_{3} & +  2.00 x_{4} & -10.00 x_{5} & + 10.00 x_{6} & -8.00 x_{7} & -4.00 x_{8} & +  9.00 x_{9} & +  8.00 x_{10} & -8.00 x_{11} & +  6.00 x_{12} & -1.00 x_{13} & -8.00 x_{14} & -6.00 x_{15} & +  3.00 x_{16}\\
 x_{23}   &  3.0 & +  7.00 x_{1} & +  2.00 x_{2} & +  9.00 x_{3} & +  4.00 x_{4} & +  7.00 x_{5} & -2.00 x_{6} & -1.00 x_{7} & -1.00 x_{8} & + 10.00 x_{9} & -7.00 x_{10} & -10.00 x_{11} & -4.00 x_{12} & +  1.00 x_{13} & -7.00 x_{14} & +  2.00 x_{15} & -4.00 x_{16}\\
 x_{24}   &  -1.0 & -10.00 x_{1} & +  1.00 x_{2} & -1.00 x_{3} & +  6.00 x_{4} & +  8.00 x_{5} & -7.00 x_{6} & -6.00 x_{7} & +  8.00 x_{8} & +  7.00 x_{9} & -7.00 x_{10} & +  3.00 x_{11} & +  5.00 x_{12} & +  2.00 x_{13} & -2.00 x_{14} & -9.00 x_{15} & +  5.00 x_{16}\\
 x_{25}   &  -1.0 & -7.00 x_{1} & -6.00 x_{2} & +  7.00 x_{3} & +  6.00 x_{4} & +  7.00 x_{5} & -9.00 x_{6} & +  8.00 x_{7} & -1.00 x_{8} & -5.00 x_{9} & -10.00 x_{10} & -8.00 x_{11} & -9.00 x_{12} & +  3.00 x_{13} & +  8.00 x_{14} & +  3.00 x_{15} & -9.00 x_{16}\\
 x_{26}   &  0.0 & -10.00 x_{1} & -1.00 x_{2} & -4.00 x_{3} & -3.00 x_{4} & +  8.00 x_{5} & +  9.00 x_{6} & +  3.00 x_{7} & +  8.00 x_{8} & -4.00 x_{9} & +  5.00 x_{10} & +  8.00 x_{11} & -7.00 x_{12} &   & -4.00 x_{14} & -1.00 x_{15} &   \\
 x_{27}   &  -3.0 & +  3.00 x_{1} & +  4.00 x_{2} & +  3.00 x_{3} & -5.00 x_{4} &   & +  4.00 x_{6} & -10.00 x_{7} & -2.00 x_{8} & +  5.00 x_{9} & -7.00 x_{10} & +  1.00 x_{11} & -10.00 x_{12} & -8.00 x_{13} & +  7.00 x_{14} & +  2.00 x_{15} & +  6.00 x_{16}\\
 x_{28}   &  2.0 & +  3.00 x_{1} & -3.00 x_{2} & +  2.00 x_{3} & +  4.00 x_{4} & +  2.00 x_{5} & -4.00 x_{6} & +  8.00 x_{7} & +  4.00 x_{8} & -6.00 x_{9} & -1.00 x_{10} & -7.00 x_{11} &   & +  3.00 x_{13} & +  6.00 x_{14} & -3.00 x_{15} & -5.00 x_{16}\\
 x_{29}   &  0.0 & +  9.00 x_{1} & -5.00 x_{2} & +  8.00 x_{3} & -10.00 x_{4} & +  2.00 x_{5} & +  5.00 x_{6} & -4.00 x_{7} & -10.00 x_{8} & +  1.00 x_{9} & +  8.00 x_{10} & -2.00 x_{11} & +  2.00 x_{12} & -9.00 x_{13} & +  2.00 x_{14} & -2.00 x_{15} & +  6.00 x_{16}\\
 x_{30}   &  2.0 & -10.00 x_{1} & +  7.00 x_{2} & -9.00 x_{3} & +  2.00 x_{4} & -6.00 x_{5} & +  4.00 x_{6} & +  1.00 x_{7} & -7.00 x_{8} & +  2.00 x_{9} & -10.00 x_{10} & +  8.00 x_{11} & +  6.00 x_{12} & -4.00 x_{13} & +  1.00 x_{14} & +  7.00 x_{15} & -8.00 x_{16}\\
 x_{31}   &  2.0 & +  8.00 x_{1} & -3.00 x_{2} & +  2.00 x_{3} & -9.00 x_{4} &   & +  5.00 x_{6} & -10.00 x_{7} & -3.00 x_{8} & -3.00 x_{9} & -1.00 x_{10} & +  6.00 x_{11} & -6.00 x_{12} & -2.00 x_{13} & +  6.00 x_{14} & + 10.00 x_{15} & +  4.00 x_{16}\\
\hline
z    &  0.0 & +  3.00 x_{1} & -5.00 x_{2} & -3.00 x_{3} & +  1.00 x_{4} & -4.00 x_{5} & +  2.00 x_{6} & +  2.00 x_{7} & +  5.00 x_{8} & -3.00 x_{9} & -1.00 x_{10} & +  1.00 x_{11} & +  3.00 x_{12} & -5.00 x_{13} & +  1.00 x_{14} & -5.00 x_{15} & +  5.00 x_{16}\\
\end{array}\]
\subsection{Initialization Phase: Dual Problem Solving}
New Objective in primal was changed to : \[ \max\ \sum_{j=1}^{16}\ - x_j \] 
Primal variable $x_j$ corresponds to dual variable $y_j$ for $j = 1,\ldots,31$
Dual Dictionary (with objective changed is): 
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 y_{1}   &  1.0 & +  6.00 y_{17} & -1.00 y_{18} & -5.00 y_{19} & -7.00 y_{20} & -6.00 y_{21} & +  7.00 y_{22} & -7.00 y_{23} & + 10.00 y_{24} & +  7.00 y_{25} & + 10.00 y_{26} & -3.00 y_{27} & -3.00 y_{28} & -9.00 y_{29} & + 10.00 y_{30} & -8.00 y_{31}\\
 y_{2}   &  1.0 & +  5.00 y_{17} & + 10.00 y_{18} & +  4.00 y_{19} & +  3.00 y_{20} & +  2.00 y_{21} & +  3.00 y_{22} & -2.00 y_{23} & -1.00 y_{24} & +  6.00 y_{25} & +  1.00 y_{26} & -4.00 y_{27} & +  3.00 y_{28} & +  5.00 y_{29} & -7.00 y_{30} & +  3.00 y_{31}\\
 y_{3}   &  1.0 & -2.00 y_{17} & +  9.00 y_{18} & -7.00 y_{19} & -3.00 y_{20} &   & +  8.00 y_{22} & -9.00 y_{23} & +  1.00 y_{24} & -7.00 y_{25} & +  4.00 y_{26} & -3.00 y_{27} & -2.00 y_{28} & -8.00 y_{29} & +  9.00 y_{30} & -2.00 y_{31}\\
 y_{4}   &  1.0 & +  7.00 y_{17} & -5.00 y_{18} & -1.00 y_{19} & -8.00 y_{20} & -4.00 y_{21} & -2.00 y_{22} & -4.00 y_{23} & -6.00 y_{24} & -6.00 y_{25} & +  3.00 y_{26} & +  5.00 y_{27} & -4.00 y_{28} & + 10.00 y_{29} & -2.00 y_{30} & +  9.00 y_{31}\\
 y_{5}   &  1.0 & +  6.00 y_{17} & -9.00 y_{18} & +  5.00 y_{19} &   & -4.00 y_{21} & + 10.00 y_{22} & -7.00 y_{23} & -8.00 y_{24} & -7.00 y_{25} & -8.00 y_{26} &   & -2.00 y_{28} & -2.00 y_{29} & +  6.00 y_{30} &   \\
 y_{6}   &  1.0 & +  8.00 y_{17} & +  5.00 y_{18} & -3.00 y_{19} & -5.00 y_{20} & +  8.00 y_{21} & -10.00 y_{22} & +  2.00 y_{23} & +  7.00 y_{24} & +  9.00 y_{25} & -9.00 y_{26} & -4.00 y_{27} & +  4.00 y_{28} & -5.00 y_{29} & -4.00 y_{30} & -5.00 y_{31}\\
 y_{7}   &  1.0  &   & +  1.00 y_{18} & -6.00 y_{19} & +  4.00 y_{20} &   & +  8.00 y_{22} & +  1.00 y_{23} & +  6.00 y_{24} & -8.00 y_{25} & -3.00 y_{26} & + 10.00 y_{27} & -8.00 y_{28} & +  4.00 y_{29} & -1.00 y_{30} & + 10.00 y_{31}\\
 y_{8}   &  1.0 & +  5.00 y_{17} & -8.00 y_{18} & -3.00 y_{19} & -1.00 y_{20} & +  7.00 y_{21} & +  4.00 y_{22} & +  1.00 y_{23} & -8.00 y_{24} & +  1.00 y_{25} & -8.00 y_{26} & +  2.00 y_{27} & -4.00 y_{28} & + 10.00 y_{29} & +  7.00 y_{30} & +  3.00 y_{31}\\
 y_{9}   &  1.0 & -2.00 y_{17} & -7.00 y_{18} & +  6.00 y_{19} & -1.00 y_{20} & +  3.00 y_{21} & -9.00 y_{22} & -10.00 y_{23} & -7.00 y_{24} & +  5.00 y_{25} & +  4.00 y_{26} & -5.00 y_{27} & +  6.00 y_{28} & -1.00 y_{29} & -2.00 y_{30} & +  3.00 y_{31}\\
 y_{10}   &  1.0 & -5.00 y_{17} & -5.00 y_{18} & -3.00 y_{19} & -5.00 y_{20} & +  9.00 y_{21} & -8.00 y_{22} & +  7.00 y_{23} & +  7.00 y_{24} & + 10.00 y_{25} & -5.00 y_{26} & +  7.00 y_{27} & +  1.00 y_{28} & -8.00 y_{29} & + 10.00 y_{30} & +  1.00 y_{31}\\
 y_{11}   &  1.0 & -3.00 y_{17} & +  7.00 y_{18} & +  4.00 y_{19} & +  8.00 y_{20} & +  1.00 y_{21} & +  8.00 y_{22} & + 10.00 y_{23} & -3.00 y_{24} & +  8.00 y_{25} & -8.00 y_{26} & -1.00 y_{27} & +  7.00 y_{28} & +  2.00 y_{29} & -8.00 y_{30} & -6.00 y_{31}\\
 y_{12}   &  1.0 & -10.00 y_{17} & -6.00 y_{18} & +  9.00 y_{19} & -10.00 y_{20} & -3.00 y_{21} & -6.00 y_{22} & +  4.00 y_{23} & -5.00 y_{24} & +  9.00 y_{25} & +  7.00 y_{26} & + 10.00 y_{27} &   & -2.00 y_{29} & -6.00 y_{30} & +  6.00 y_{31}\\
 y_{13}   &  1.0 & -2.00 y_{17} & -4.00 y_{18} & +  9.00 y_{19} & -8.00 y_{20} & +  6.00 y_{21} & +  1.00 y_{22} & -1.00 y_{23} & -2.00 y_{24} & -3.00 y_{25} &   & +  8.00 y_{27} & -3.00 y_{28} & +  9.00 y_{29} & +  4.00 y_{30} & +  2.00 y_{31}\\
 y_{14}   &  1.0 & +  2.00 y_{17} & +  6.00 y_{18} & -1.00 y_{19} & +  9.00 y_{20} &   & +  8.00 y_{22} & +  7.00 y_{23} & +  2.00 y_{24} & -8.00 y_{25} & +  4.00 y_{26} & -7.00 y_{27} & -6.00 y_{28} & -2.00 y_{29} & -1.00 y_{30} & -6.00 y_{31}\\
 y_{15}   &  1.0 & +  9.00 y_{17} & -8.00 y_{18} & -8.00 y_{19} & -8.00 y_{20} & -3.00 y_{21} & +  6.00 y_{22} & -2.00 y_{23} & +  9.00 y_{24} & -3.00 y_{25} & +  1.00 y_{26} & -2.00 y_{27} & +  3.00 y_{28} & +  2.00 y_{29} & -7.00 y_{30} & -10.00 y_{31}\\
 y_{16}   &  1.0 & +  5.00 y_{17} & -3.00 y_{18} & -9.00 y_{19} & -1.00 y_{20} & -9.00 y_{21} & -3.00 y_{22} & +  4.00 y_{23} & -5.00 y_{24} & +  9.00 y_{25} &   & -6.00 y_{27} & +  5.00 y_{28} & -6.00 y_{29} & +  8.00 y_{30} & -4.00 y_{31}\\
\hline
z    &  -0 & +  3.00 y_{17} & -3.00 y_{18} & +  1.00 y_{19} &   & -3.00 y_{21} & +  3.00 y_{22} & -3.00 y_{23} & +  1.00 y_{24} & +  1.00 y_{25} &   & +  3.00 y_{27} & -2.00 y_{28} &   & -2.00 y_{30} & -2.00 y_{31}\\
\end{array}\]
Unbounded Dictionary!
Initialization returns unbounded dual dictionary after 17 pivots
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 y_{1}   &  111.663444889 & -11.84 y_{3} & -374.98 y_{18} & -6.21 y_{10} & -54.62 y_{20} & -269.86 y_{24} & -16.50 y_{5} & -507.19 y_{23} & -2.41 y_{12} & -39.71 y_{9} & -23.35 y_{7} & -10.65 y_{16} & + 51.29 y_{28} & -201.45 y_{29} & + 245.64 y_{30} & + 298.97 y_{31}\\
 y_{2}   &  79.2171393332 & -8.45 y_{3} & -247.20 y_{18} & -3.94 y_{10} & -28.62 y_{20} & -196.32 y_{24} & -11.12 y_{5} & -351.49 y_{23} & -1.88 y_{12} & -27.93 y_{9} & -17.01 y_{7} & -7.90 y_{16} & + 38.83 y_{28} & -127.38 y_{29} & + 154.18 y_{30} & + 223.55 y_{31}\\
 y_{27}   &  1.95869473718 & -0.28 y_{3} & -6.26 y_{18} & -0.12 y_{10} & -1.36 y_{20} & -5.43 y_{24} & -0.32 y_{5} & -9.47 y_{23} & -0.04 y_{12} & -0.69 y_{9} & -0.33 y_{7} & -0.20 y_{16} & +  1.45 y_{28} & -4.44 y_{29} & +  5.24 y_{30} & +  4.32 y_{31}\\
 y_{21}   &  4.13568494464 & -0.44 y_{3} & -13.92 y_{18} & -0.18 y_{10} & -1.95 y_{20} & -11.06 y_{24} & -0.62 y_{5} & -18.20 y_{23} & -0.13 y_{12} & -1.44 y_{9} & -0.87 y_{7} & -0.46 y_{16} & +  2.07 y_{28} & -7.14 y_{29} & +  8.55 y_{30} & + 11.30 y_{31}\\
 y_{8}   &  43.4625767206 & -4.74 y_{3} & -147.07 y_{18} & -1.62 y_{10} & -21.37 y_{20} & -122.09 y_{24} & -5.92 y_{5} & -193.06 y_{23} & -1.43 y_{12} & -15.36 y_{9} & -8.99 y_{7} & -4.41 y_{16} & + 18.62 y_{28} & -61.37 y_{29} & + 88.29 y_{30} & + 122.07 y_{31}\\
 y_{6}   &  25.147450105 & -3.47 y_{3} & -65.22 y_{18} & -0.69 y_{10} & -23.60 y_{20} & -57.79 y_{24} & -3.56 y_{5} & -109.83 y_{23} & -1.40 y_{12} & -7.97 y_{9} & -4.86 y_{7} & -2.19 y_{16} & + 10.55 y_{28} & -49.93 y_{29} & + 43.90 y_{30} & + 60.87 y_{31}\\
 y_{26}   &  5.01742330602 & -0.55 y_{3} & -17.09 y_{18} & -0.32 y_{10} & -2.60 y_{20} & -12.79 y_{24} & -0.79 y_{5} & -22.16 y_{23} & -0.11 y_{12} & -1.73 y_{9} & -1.00 y_{7} & -0.51 y_{16} & +  2.56 y_{28} & -9.53 y_{29} & + 11.80 y_{30} & + 13.09 y_{31}\\
 y_{19}   &  0.951291064179 & -0.12 y_{3} & -2.54 y_{18} & -0.06 y_{10} & -0.24 y_{20} & -1.79 y_{24} & -0.10 y_{5} & -3.85 y_{23} & -0.00 y_{12} & -0.32 y_{9} & -0.22 y_{7} & -0.12 y_{16} & +  0.36 y_{28} & -1.81 y_{29} & +  2.38 y_{30} & +  2.56 y_{31}\\
 y_{25}   &  4.09773554296 & -0.42 y_{3} & -13.56 y_{18} & -0.17 y_{10} & -1.00 y_{20} & -10.39 y_{24} & -0.58 y_{5} & -19.10 y_{23} & -0.07 y_{12} & -1.52 y_{9} & -0.94 y_{7} & -0.40 y_{16} & +  1.79 y_{28} & -6.20 y_{29} & +  7.75 y_{30} & + 12.09 y_{31}\\
 y_{4}   &  20.3453773163 & -3.42 y_{3} & -64.09 y_{18} & -2.08 y_{10} & -35.63 y_{20} & -55.92 y_{24} & -3.54 y_{5} & -89.80 y_{23} & -0.81 y_{12} & -5.69 y_{9} & -2.12 y_{7} & -1.69 y_{16} & +  9.76 y_{28} & -50.03 y_{29} & + 65.99 y_{30} & + 40.60 y_{31}\\
 y_{11}   &  13.3715415783 & -0.26 y_{3} & -34.15 y_{18} & +  0.57 y_{10} & + 23.99 y_{20} & -31.72 y_{24} & -0.56 y_{5} & -52.38 y_{23} & +  0.25 y_{12} & -6.17 y_{9} & -4.76 y_{7} & -1.43 y_{16} & + 10.85 y_{28} & +  8.13 y_{29} & -12.12 y_{30} & + 51.85 y_{31}\\
 y_{17}   &  6.4026880188 & -0.79 y_{3} & -20.64 y_{18} & -0.39 y_{10} & -4.14 y_{20} & -16.24 y_{24} & -0.96 y_{5} & -28.62 y_{23} & -0.20 y_{12} & -2.20 y_{9} & -1.24 y_{7} & -0.62 y_{16} & +  3.23 y_{28} & -12.72 y_{29} & + 14.93 y_{30} & + 16.51 y_{31}\\
 y_{13}   &  29.0660736358 & -3.41 y_{3} & -92.71 y_{18} & -1.50 y_{10} & -22.40 y_{20} & -74.78 y_{24} & -4.03 y_{5} & -124.63 y_{23} & -0.56 y_{12} & -9.66 y_{9} & -5.50 y_{7} & -3.41 y_{16} & + 14.69 y_{28} & -47.64 y_{29} & + 73.04 y_{30} & + 70.29 y_{31}\\
 y_{14}   &  19.3859839686 & -1.37 y_{3} & -62.21 y_{18} & -1.25 y_{10} & +  0.44 y_{20} & -42.59 y_{24} & -2.62 y_{5} & -64.59 y_{23} & -0.60 y_{12} & -6.44 y_{9} & -4.07 y_{7} & -2.04 y_{16} & +  3.51 y_{28} & -31.78 y_{29} & + 35.30 y_{30} & + 48.01 y_{31}\\
 y_{15}   &  52.1298509809 & -5.87 y_{3} & -180.61 y_{18} & -3.11 y_{10} & -40.12 y_{20} & -123.34 y_{24} & -7.77 y_{5} & -231.62 y_{23} & -1.63 y_{12} & -17.98 y_{9} & -10.01 y_{7} & -4.77 y_{16} & + 30.48 y_{28} & -95.17 y_{29} & + 105.97 y_{30} & + 126.06 y_{31}\\
 y_{22}   &  4.11936915938 & -0.38 y_{3} & -14.18 y_{18} & -0.18 y_{10} & -0.96 y_{20} & -10.48 y_{24} & -0.56 y_{5} & -18.58 y_{23} & -0.07 y_{12} & -1.55 y_{9} & -0.95 y_{7} & -0.43 y_{16} & +  2.20 y_{28} & -6.09 y_{29} & +  7.53 y_{30} & + 12.27 y_{31}\\
\hline
z    &  30.0842275193 & -3.57 y_{3} & -100.60 y_{18} & -1.75 y_{10} & -14.78 y_{20} & -74.49 y_{24} & -4.34 y_{5} & -141.35 y_{23} & -0.60 y_{12} & -10.82 y_{9} & -6.10 y_{7} & -2.90 y_{16} & + 14.59 y_{28} & -56.31 y_{29} & + 65.59 y_{30} & + 78.05 y_{31}\\
\end{array}\]
Original Problem is Infeasible
Infeasible Problem.. Early Exit
\end{document}
